integral of x/(sqrt(x+5))

解答

\int \frac{x}{x + 5} d x

2 (\frac{1}{3} (x + 5)^{\frac{3}{2}} - 5 x + 5) + C

求解步骤

\int \frac{x}{x + 5} d x

使用换元积分法

= \int 2 (u^{2} - 5) d u

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int u^{2} - 5 d u

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int u^{2} d u - \int 5 d u)

\int u^{2} d u = \frac{u ^{3}}{3}

\int 5 d u = 5 u

= 2 (\frac{u ^{3}}{3} - 5 u)

u = x + 5

代回

= 2 (\frac{( x + 5 ) ^{3}}{3} - 5 x + 5)

化简

2 (\frac{( x + 5 ) ^{3}}{3} - 5 x + 5) : 2 (\frac{1}{3} (x + 5)^{\frac{3}{2}} - 5 x + 5)

= 2 (\frac{1}{3} (x + 5)^{\frac{3}{2}} - 5 x + 5)

解答补常数

= 2 (\frac{1}{3} (x + 5)^{\frac{3}{2}} - 5 x + 5) + C

\int \frac{4}{t} d t

\frac{\partial}{\partial x} (2 x y)

d er i v a t i v e f (g (x))

\frac{\partial}{\partial x} (x^{y^{z}})

t an g e n t f (x) = 7 x + 1, at x = 5